A-priori estimates for some classes of elliptic problems

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Università degli Studi di Catania Dipartimento di Matematica e Informatica

Tesi di Dottorato in Matematica

A-priori estimates for some classes of

elliptic problems

Candidata Greta Marino

Relatore

Prof. Salvatore A. Marano Correlatore

Prof. Sunra J.N. Mosconi

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”The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge.” Stephen Hawking

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Acknowledgements

I would like to express my sincere gratitude to every people who supported me during the years of my PhD. Above all I am deeply grateful to my family, for their silent and tireless presence, and for sharing with me all the good moments as well as the less good ones.

I also want to thank my supervisor, Prof. S.A. Marano, for all the opportunities he gave me during these years, his encouragements and his conﬁdence in my capacities. I would like to thank all my friends, both the people that are sharing my way since the years of the school and the people who entered in my life during the course of the time: not in order of importance, my aﬀection is extended to Mimma, Simona, Silvia, Concia, Concy, Andrea, Lorenzo, Nicola, Giorgia, Guido, Patrick.

My last thanks is addressed to Prof. Sunra Mosconi. I am completely grateful to him, because he showed me the way to become a good mathematician and with his example contributed to build the person who I am.

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Introduction

This thesis arises with the purpose of emphasizing some aspects of a powerful tool widely employed in mathematical analysis: the a-priori estimates. Indeed, it is well known that a-priori estimates play an important role in the theory of partial diﬀerential equations and in the calculus of variations, since they are intimately related with the existence of solutions for a given problem. We will present three of the papers written during this PhD, which are connected by this topic.

Chapter 1 contains some results obtained jointly with Prof. S. Mosconi from the University of Catania and published in the paper [92] on Journal of Diﬀerential Equations. The existence of solutions to the equation

u0000+ qu00+ F0(u) = 0, (1)

where q ∈ R and F is a C2_{, coercive, and quasi-convex function, that is,}

F0(t)t ≥ 0, ∀ t ∈ R.

is investigated. Equation (1) can be classiﬁed according to the sign of q. More pre-cisely, we say that (1) is the Extended Fisher-Kolmogorov (EFK) equation if q ≤ 0, while (1) is the Swift-Hohenberg (S-H) equation if q > 0.

Equation (1) has important applications in the real world, since it describes complex patterns in many systems that come from Physics and Mechanics. More precisely, the EFK equation arises as a model equation for certain physical systems that are

bistable and in the study of phase transitions near singular points (see [41] and [117],

respectively). On the other hand, the S-H equation has been widely employed in the description of cellular ﬂows and in the context of lasers (see [130, 30] as well as [81, 133]). But the greatest application of the S-H equation is in the description of the suspension bridges. The idea was proposed by Lazer, McKenna and Walter [77, 95, 96] in order to model a suspension bridge as a vibrating beam supported by cables, with a nonlinear response to loading and a constant weight per unit length due to gravity. All these topics represent the ﬁrst part of Chapter 1.

The second part focuses instead on our results [92], which provide an answer to some questions posed in [79]. We obtained some nonexistence results for the EFK equa-tion, see Theorems 1.6.1 and 1.6.3, and some existence results for the S-H equaequa-tion, which also give the exact parameter range for which the equation has a nontrivial bounded solution, see Theorems 1.6.4 and 1.6.5.

The last part of Chapter 1 deals with the asymptotic behavior of the periodic solu-tions obtained in the aforementioned results. More precisely, we show that, under suitable conditions on the function F , every solution of equation (1) is such that kuk∞→ 0, as q → 0; see Corollary 1.6.1 and Theorem 1.6.7, respectively.

Here the importance of the a-priori estimates lies in the fact that they allow us to obtain some qualitative and global properties of solutions.

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Chapter 2 is based on the results obtained in [93] written in collaboration with Prof. P. Winkert from the University of Technology Berlin, Germany, and already published on Nonlinear Analysis. In this work, we studied the global boundedness of solutions to the following boundary value problem

− div A(x, u, ∇u) = B(x, u, ∇u) in Ω,

A(x, u, ∇u) · ν = C(x, u) on ∂Ω, (2)

where ν(x) denotes the outer unit normal of Ω at x ∈ ∂Ω, and A, B, and C satisfy suitable p-structure conditions which can allow critical growth, in Ω as well as on

∂Ω.

Boundary value problems with critical exponent have always represented an im-portant task to overcome. Indeed, since the embedding W1,p_{(Ω) ,→ L}p∗

(Ω) is not compact, this in turn implies that the functional associated to a prescribed prob-lem does not satisfy the Palais-Smale condition. Hence, there are serious difficulties when trying to find its critical points by standard variational methods. Several au-thors have developed different strategies in order to overcome these difficulties, and a detailed description of them can be found in Section 2.2.

The main motivation for studying critical problems like (2) stems from the fact that they arise from some variational problems in Geometry and Physics where lack of compactness also occurs. In particular, (2) can be seen as a generalization of the classical Yamabe problem

− ∆u = f(x)u + h(x)uNN+2−2, (3) where f and h are smooth functions. It is well known that there is no stable regularity theory for solutions of (3), which reﬂects the diﬃculty of the Yamabe problem. Nevertheless, it was proven by Trudinger [137] that any W1,2_{(Ω) solution of}

(3) is in fact smooth, but its regularity estimate depends on the solution itself. In this spirit, the main result of Chapter 2, Theorem 2.6.1, can be seen as a generalization of Trudinger’s work. Theorem 2.6.1, whose proof is performed in Subsection 2.6.2, contains two important assertions. First of all, it states that every solution u ∈

W1,p_{(Ω) of problem (2) is in L}r_{(Ω) for every r < +∞, and then that u is actually} in L∞_{(Ω), with a bound which depends on the given data and on the solution}

itself. The main tool which allowed us to obtain these results is a modiﬁed version of Moser’s iteration technique, which in turn is based on the books [44, 129], and which is also brieﬂy presented in Section 2.5.

Chapter 2 ends with some regularity results. We proved that, with some additional assumptions on the functions A and C, every weak solution of (2) is actually in

C1,β(Ω), for some β ∈ (0, 1), see Theorem 2.6.2.

The importance of the a-priori estimates is emphasized not only in Theorem 2.6.1, but rather in the fact that the a-priori bound of a given solution to problem (2) directly entails regularity considerations ensuring that u ∈ C1,β_(Ω).

Chapter 3 is based on the results obtained in [91], written in collaboration with Prof. S.A. Marano and Prof. A. Moussaoui from the University of Mira Bejaia, Algeria. It focuses on an existence result for the following singular system

−∆p1u = a1(x)f(u, v) in R N_, −∆p2v = a2(x)g(u, v) in R N_, u, v > 0, u, v_{→ 0 as |x| → +∞.} (4)

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Here with the word singular we mean an equation which contains terms that could go to +∞ as variables approach zero. And system (4) is eﬀectively singular since, according to our hypotheses, the functions f and g have a growth that involves also negative exponents.

Singular semilinear systems in a bounded domain were introduced by Gierer and Meinhardt [57] as a mathematical model in biochemical processes. The non singular version has instead been widely employed in the study of interacting populations. These interactions, whose description is contained in the first part of the chapter, are of three types: the predator-prey model, which occurs if the growth rate of one population is decreasing while the other is increasing with respect to the first; the competition model, which occurs if the growth rate of each population is decreasing with respect to the other; finally, the mutualism, which happens when each popu-lation’s growth rate is enhancing. All these situations are described by systems of ordinary differential equations or partial differential equations that also take into account the tendency of each population to spatially diffuse.

The second part of the chapter treats singular systems, starting from the semilin-ear case in a bounded domain, see Subsection 3.4.1, and then concentrating on the quasilinear case, see Subsection 3.4.2. Since variational methods do not work, differ-ent other techniques, mainly based on fixed point argumdiffer-ents in a sub-supersolution setting, were developed. Of course these problems can be generalized considering the case when Ω = RN_{: the semilinear case was treated for example in [101], while} the quasilinear case, to the best of our knowledge, has never been studied in the literature until our work [91], which is contained in Section 3.5.

Here we assume some structure conditions on the functions f and g which do not al-low to reduce system (4) to the Gerier-Meinhardt’s type. The main idea for solving (4) is to perturb the system with a parameter ε > 0, and then to apply Schauder’s ﬁxed point theorem in order to get a solution (uε, vε), for every ε > 0. Finally, letting ε → 0+ _{yields a weak solution (u, v) of (4).}

Here, the a-priori estimates play a crucial role, because it is only once we know that a solution (u, v), as well as (uε, vε), is uniformly bounded that we can apply some comparisons which directly lead to the existence results.

All the chapters are structured in a similar way. We ﬁrst introduce the concrete problems which inspired our works. Then, after a description of some known results from the literature, we present the results obtained in our papers. Finally, some open problems and further possible developments are listed.

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Chapter 1

Model equations in physical

pattern formation

1.1 Introduction

One of the most interesting aspects of the complex dynamics that governs natural phenomena is the occurrence of instabilities and symmetry breakings that lead to the formation of coherent spatio-temporal structures on macroscopic scales. When a physico-chemical system is maintained far from its thermal equilibrium by the application of external constraints, it may go through a succession of instabilities that induce various types of spatio-temporal pattern formation. These phenomena are studied using the methods of nonlinear dynamics and instability theory, because such a organization seems to be related to technological problems that arise, among others, in physics, chemistry and nonlinear optics. Since these systems are described by nonlinear partial differential equations, it is very difficult to solve these problems because of the impossibility to obtain analytical solutions. However, in the 1970s general techniques, based on the analogies between phase transitions and critical phenomena, were developed. In particular, the fact that similar phenomena appear in very different systems (such as spiral waves in chemical systems, cardiac activity, hydrodynamical instabilities in liquid crystals) shows that they are not induced by the microscopic properties of the system but they are triggered by collective effects including a large number of individuals (atoms, molecules, cells).

In order to achieve a better understanding of the dynamical behavior of systems far from equilibrium, well-chosen model equations have been proposed. Even if they are often simpler than the full equations describing those systems, they allow us to underline the mechanisms that are responsible for the formation and evolution of complex patterns. Classical model equations are typically second-order PDEs. One of the most important is the widely studied Fisher-Kolmogorov equation, a nonlinear second-order diﬀusion equation proposed in 1937 as a model for the interaction of dispersal and ﬁtness in biological populations.

Our interest is instead concentrated in a series of higher-order partial diﬀerential equations that have been taken as model in the study of pattern formation in systems from physics and mechanics. For special classes of solutions such as stationary solutions or traveling waves, many of these equations reduce to a simpler one, of the form

d4_u

dx4 + q

d2_u

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Here q is a real-valued eigenvalue parameter which measures the tendency of the equation to form complex patterns, and f is a given function. Equation (1.1.1), and the properties of its bounded solutions for diﬀerent values of q and functions f, are the core of this chapter.

1.2 Model equations

Well-chosen model equations have always played an important role in applied math-ematics, because of their presence in a great variety of physical, chemical and bio-logical systems. Classical examples are the heat equation, the wave equation and the Laplace equation, which describe processes like diﬀusion, dispersion and wave propagation, and also give their mutual interactions and their quantitative descrip-tion. They are typically linear second-order partial diﬀerential equations. However, since many problems in the sciences and in engineering are intrinsecally nonlinear, it became necessary to introduce nonlinear generalizations.

1.2.1 Second-order model equations

A very well-known example from the literature is the Fisher-Kolmogorov (FK)

equa-tion ∂u ∂t = ∂2_u ∂x2 + u − u 3_. _(1.2.1)

It was originally proposed to study the fronts that arise in population dynamics. A front is a propagating interface between two different steady states and can be viewed as a balance between diffusive forces coupling different points in the field, and reactive forces which move the system from unstable to stable states. At first sight, front propagation through unstable states might seem to be an esoteric subject. In reality, however, there are many examples where this phenomenon is an essential element of the dynamics. For example, fronts naturally arise in convectively unstable systems, in which a state is unstable, but perturbations are convected away faster than they grow out.

From a qualitative point of view, equation (1.2.1) exhibits an unstable uniform state, u = 0, and two stable uniform states, u = ±1. Fronts that oscillate around zero exist, but they are considered unphysical, since they represent negative popu-lation densities. Consequently, interest is focused on strictly non-negative solutions. Fronts in the FK equation connect the unstable to the stable state, and move in such a way as to destabilize the unstable state.

The FK equation is often called the real Ginzburg-Landau (GL) equation, since it is the real version of the complex GL equation, an envelope equation that describes the dynamics of wave envelopes near transition in hydrodynamic systems.

Another second-order model equation of interest is the following, known as the

sine-Gordon (sG) equation

∂2u ∂t2 =

∂2u

∂x2 − sin u.

It is a nonlinear hyperbolic partial diﬀerential equation in 2 dimensions involving the d’Alembert operator and the sine of the unknown function. It was originally introduced by E. Bour [21] in the study of surfaces of constant negative curvature and rediscovered by Frenkel and Kontorova [50] in their study of crystal dislocations. It

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attracted a lot of attention in the 1970s because of the presence of soliton solutions, and was also widely used in studies of nonlinear wave propagation, see [45, 119]. 1.2.2 Higher-order model equations

The equations introduced above are not able to describe complex patterns for which equation (1.1.1) plays an important role. A typical example of such a pattern is the phenomenon of localized buckling in mechanics, which happens when the deﬂections are conﬁned to a small portion of the otherwise unperturbed material. To understand the formation of complex spatial and temporal patterns, higher-order scalar model equations and systems of equations have been proposed. Below we list some of these equations that are of interest for us, beginning with the one that can be considered their prototype.

Elastic beam

The mechanics of solid bodies, regarded as continuous media, forms the content of the theory of elasticity, whose basic equations were established in the 1820s by Cauchy and Poisson. Under the action of applied forces, solid bodies exhibit defor-mation to some extent, and so change in shape and volume. The three-dimensional equations of a continuous solid elastic medium vibrations are of a great complexity and in general cannot be solved analytically. However, elastic solids present geo-metrical characteristics which simplify the mathematical analysis of their vibrations. These simplifications have led to the theories of beams, plates and shells. In par-ticular, the theory of beams consists of constructing one-dimensional models and in this sense represents the simplest continuous media. This simplicity is extremely useful since it leads to obtain analytical solutions of the problem equations and, con-sequently, to study the vibratory phenomena in a comprehensive fashion. Research of the basic vibratory phenomena results in the identification of three elementary movements: longitudinal vibrations, vibrations of torsion and bending vibrations. The study of coupled longitudinal movements, torsion and bending is possible, but with an increased difficulty of resolution. Probably one of the simplest equation for an elastic beam is the following

∂2_u

∂t2 + κ 2∂4u

∂x4 = 0,

where κ is characteristic of the given bar.

The Extended Fisher-Kolmogorov (EFK) equation

This equation, which can be regarded as a natural extension of the FK equation (1.2.1), was proposed in 1988 by Dee and van Saarloos [41] during their studies on wave propagation, and models physical systems that are bistable

∂u ∂t = −γ ∂4_u ∂x4 + ∂2_u ∂x2 + u − u 3_, _{γ > 0.} _(1.2.2)

Indeed, as in the FK equation, the EFK equation has two uniform states u(x) = ±1 which are stable, separated by a third uniform state u(x) = 0 which is unstable. The choice γ > 0 is dictated by the physical requirement that the model is stable at short wavelengths, but otherwise the fourth-order spatial derivative does not dramatically

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alter the qualitative features of the homogeneous states. Indeed, the u = 0 state remains unstable to long-wavelength perturbations, while the other states remain absolutely stable.

The EFK equation also arises in the study of phase transitions near singular points, the so-called Lifshitz points (LP for short). It can occur in a variety of diﬀerent systems, including magnetic compounds and alloys, liquid crystals and charge-transfer salts. Finally, equation (1.2.2) arises as amplitude equation at the onset of instability near certain degenerate states. In [117] it was shown, through numerical simulations of the full reaction-diﬀusion system, that the behavior of small perturbations near the degeneration is best described by the EFK equation, as opposed to the classical Ginzburg-Landau model.

The Swift-Hohenberg (S-H) equation

It was first proposed by Swift and Hohenberg in order to study the effects of thermal fluctuations on a fluid near the Rayleigh-Bénard instability [130]

∂u ∂t = − 1 + ∂2 ∂x2 2 u + κu_{− u}3, κ_{∈ R.} (1.2.3) Consider a horizontal layer of fluid in which an adverse temperature gradient is main-tained by heating the underside. We say that the temperature gradient is adverse because, on account of thermal expansion, the fluid at the bottom will be lighter than the fluid at the top; and this top-heavy arrangement is potentially unstable. Because of this instability there will be a tendency on part of the fluid to redistribute itself and remedy the weakness in its arrangement. However, this natural tendency will be inhibited by its own viscosity. In other words, we expect that the adverse temperature gradient must exceed a certain value before the instability can manifest itself. The earliest experiments to demonstrate in a definitive manner the onset of thermal instability in fluids are those of Bénard in 1900, though the phenomenon of thermal convection itself had been recognized earlier by Count Rumford (1797) and James Thomson (1882). The principal facts they established are the following; first, a certain critical adverse temperature gradient must be exceed before insta-bility can set in; and second, the motions which ensue on surpassing the critical temperature gradient have a stationary cellular character. In a fundamental paper [116], Lord Rayleigh showed that what decides the stability or otherwise of a layer of fluid heated from below is the numerical value of the non-dimensional parameter

R = gαβ κν d

4_,

often called the Rayleigh number. Here g denotes the acceleration due to gravity, d the depth of the layer, β the uniform adverse temperature gradient, and α, κ and ν are the coeﬃcients of volume expansion, thermometric conductivity and kinematic viscosity, respectively. Rayleigh further showed that instability must set in when R exceeds a certain critical value Rc; and, when it happens, a stationary pattern of motions must come to prevail.

Equation (1.2.3) has also been used by Pomeau and Manneville [111] to study the phenomenon of wavenumber selection in cellular ﬂows, which follows from the breaking of translational invariance in large but ﬁnite structures. Finally, several authors adapted this equation in the context of lasers, see [81, 133].

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The Suspension Bridge equation

This equation was proposed by Lazer, McKenna and Walter [77, 95, 96] and models a suspension bridge as a vibrating beam supported by cables, which has a non-linear response to loading and a constant weight per unit length due to gravity. The unknown u(x, t) measures deﬂection from the unloaded state and is therefore applicable to vertical oscillations

∂2u ∂t2 +

∂4u

∂x4 + (u + 1)+− 1 = 0. (1.2.4)

As written in [77], much of interest in this ﬁeld started after the breakdown of the Tacoma Narrows suspension bridge [5], which was ﬁrst subject to large-scale oscillations, followed by the collapse of the structure.1 _{The standard explanation}

of this phenomenon was that the bridge behaves like a particle of mass one at the end of the spring, with spring constant k, which is subject to a forcing term of frequency µ/2π. If µ is very close to √k, then large oscillations result. If µ is not,

then it does not. Accordingly, the forcing term came from a train of alternating vortices being shed by the bridge as the wind blew past it. The frequency just happened to be at a value very close to the resonant frequency of the bridge. Thus, even though the magnitude of forcing term was small, the phenomenon of linear resonance was enough to explain the large oscillation and the eventual collapse of the bridge. Anyway, this explanation was not persuasive.

As made clear in [5, 18], suspension bridges have a history of large-scale oscilla-tion and catastrophic failure under high and even moderate winds. Earlier bridges, such as the Bronx-Whitestone Bridge or the Golden Gate Bridge, had shown oscil-latory behavior due to the action of wind. What distinguished the Tacoma Narrows was the extreme flexibility of its roadbed. This resulted a pronounced tendency to oscillate vertically, under widely differing wind conditions. The bridge was also affected by another type of oscillation just prior to the collapse: a pronounced tor-sional mode observed after the bridge went into large vertical motion. Furthermore, a wind-tunnel study of a scale model of the Tacoma Narrows Bridge in a variety of wind conditions showed that, when attempting to model large amplitude oscillation, the behavior is almost perfectly linear, see [5, Appendix VIII].

Thus, several interesting questions from the mathematical point of view arise. For example, to understand what in the nature of suspension bridges makes them so prone to large-scale oscillation; to ﬁnd an explanation of the fact that the bridge would go into large oscillation under the impulse of a single gust, or would re-main motionless in strong winds; to explain how the large vertical oscillations could rapidly change to torsional; to study the existence of the traveling waves; to get a formal description of why the motion is linear over small to medium range oscilla-tion. It is important to observe that the current explanation of these phenomena was highly incorrect until the work of Lazer and McKenna [77], who constructed the right mathematical model, proving that what distinguishes suspension bridges is their fundamental nonlinearity. The restoring force due to a cable is such that it strongly resists expansion, but does not resist compression. Thus, the simplest func-tion to model this force would be a constant times u, the expansion, if u is positive, but zero if u is negative, corresponding to compression, see equation (1.2.3).

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Water waves

The behavior of steady nonlinear water waves on the surface of an inviscid heavy ﬂuid layer has received much attention during the past century, both from the math-ematical and from the physical side. Some interesting problems are, for example, the existence of solitary and cnoidal (periodic) waves in the presence of surface ten-sion or the description of the reaction of a ﬂuid to a localized pressure distribution moving over its surface with constant speed.

In its long history, the analysis of nonlinear surface waves has been promoted by scientists of various backgrounds, and a vast literature is available for the unforced case, which happens when the pressure at the surface is constant, see [126, 148, 152, 153]. On the other hand, the resonant case, which occurs when the pressure speed coincides with the critical wave speed, became of particular importance and diﬃculty.

In [73], nonlinearly resonant water waves are analyzed with the only assumption of moderate wave amplitudes. They reduced to waves in two dimensions which, after a suitable rescaling, lead to the following single fourth-order ordinary diﬀerential equation

u0000+ P u00+ u − u2 = 0. (1.2.5)

Here the function u(x) is related to η(x) − 1, where η is the dimensionless depth of the water, while the coeﬃcient P is a negative constant.

Equation (1.2.5) has been the object of much recent study (see [6, 26, 66, 67] and the references therein). In particular, in [6] the existence of homoclinic orbits connecting the zero equilibrium of (1.2.5) is studied, and they can be interpreted as indicating the presence of spatially localized buckling.

The nonlinear Schr¨odinger equation

It is well-known that the dynamics of optical fibers is governed by the nonlinear Schrödinger equation (NLSE). By contrast to the usual theory of evolutionary PDEs, in the NLSE the evolution variable is the ’space’ variable, namely the longitudinal coordinate of the fiber. In [2] a generalized NLSE with a negative fourth-order dispersion term is investigated. It can be obtained under the usual assumptions of the absence of derivative nonlinearities which, after a suitable rescaling, lead to the following equation i∂v ∂x+ ∂2v ∂t2 − ∂4v ∂t4 + |v| 2_{v = 0.} _(1.2.6)

Stationary pulse-like solutions of equation (1.2.6) have the form

v(x, t) = u(k, t)eikx, (1.2.7)

where k is the soliton propagation constant and u(s, t) is a real function of its parameters. Substitution of the ansatz (1.2.7) into equation (1.2.6) gives

u0000− u00+ ku − u3_{= 0,} _(1.2.8)

where k is now the only parameter of the problem. Equation (1.2.8) is a nonlinear ordinary diﬀerential equation of fourth order, and its localized solutions give the stationary soliton-like solutions of equation (1.2.6). From the mathematical point of view, the treatment of the Schr¨odinger equation (even the linear one) may be

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delicate since such equation possesses a mixture of the properties of parabolic and hyperbolic equation. In the NLSE for optical ﬁbers, because of attenuation, there is not conservation of energy.

1.3 The reduction to an ODE

When we look for special classes of solutions, such as stationary solutions or traveling waves, all the aforementioned equations reduce to an ordinary diﬀerential equation of the type (1.1.1), which can be written in the equivalent form

u0000+ qu00+ F0(u) = 0. (1.3.1)

Here with primes we mean the diﬀerentiation with respect to the variable x. The function F is often called the potential although, according to the usual terminology in classical mechanics, its opposite −F should be called the potential.

After a suitable rescaling, we can write the stationary version of all the equations (1.2.2)-(1.2.5), (1.2.8) in the form (1.3.1). More precisely, for q = −1/√γ and

F (u) = 1₄_{(1 − u}2₎2_{, we obtain the Extended Fisher-Kolmogorov equation}

u0000+ qu00+ u3_{− u = 0.} (1.3.2)

When k > 1, for q = 2/√k− 1 and F (u) = 1

4(1−u2)2, we have the Swift-Hohenberg

equation. Traveling wave solutions u(x, t) = w(x − ct) of equation (1.2.4) lead to equation (1.3.1) for q = c2_{. When q = P and F (u) =} u2

2 −u

3

3 , we have the water

waves equation (1.2.5), while for q = −1/√k and F (u) =₋₍u₂2 _{− k}2) we have the

nonlinear Schr¨odinger equation (1.2.8).

Equation (1.3.1) can be classiﬁed according to the sign of the parameter q. When

q _{≤ 0 we say that it is of the Extended Fisher-Kolmogorov-type, while it is of the}

Swift-Hohenberg-type if q > 0. We can further classify equation (1.3.1) according to the choice of the potential F . There are several possibilities.

1. F is a double-well coercive potential. In this case F exhibits two zeros, or

equilibria, u(x) = x1,2 at the same energy level. Potentials of this type appear, for

example, in the EFK equation or in the S-H equation when k > 1. When F is a multi-well potential, equation (1.3.1) was proposed in [59] as a model for ternary mixtures in order to overcome the defects of the classical Ginzburg-Landau models, which rules out observed transitions between non consecutive equilibria.

2. F is a single-well coercive potential. It has only one zero and goes to +_∞

both at +∞ and −∞. It appears in equation (1.2.4) or in (1.2.3) when k < 1 (see [106]), and in general in model equations for suspension bridges (see [31]).

3. F is an anticoercive potential. It is such that its opposite _{−F is coercive. We}

can observe it, for example, in the nonlinear Schr¨odinger equation (1.2.6).

4. F is one-sided coercive potential. It is such that F (t) _{→ +∞ for t → +∞}

and F (t) → −∞ for t → −∞, or vice-versa. Such a potential is typical of the water waves equation (1.2.5).

Both of these classiﬁcations can be mixed, and thus one can obtain an EFK equation with a single-well coercive potential, and so on.

There are two important functionals associated to (1.3.1). First, when we mul-tiply the equation by u0 _{and integrate, we obtain the energy or Hamiltonian}

E(u) := u0u000−1₂u002+q

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The energy is a conserved quantity along orbits of (1.3.1). This means that, if u is a solution of (1.3.1), then u0u000₋1 2u002+ q 2u02+ F (u) := constant := E. (1.3.4) Second, the Lagrangian action associated with this Hamiltonian

J (u) := ˆ 1 2|u00(x)|2− q 2|u0(x)|2+ F (u(x)) dx.

The solutions of (1.3.1) correspond to critical points of the action J (u) and vice-versa. The domain of integration depends on the type of solution under investiga-tion. We will go into more details of this variational structure in Section 1.4.3.

In many physical problems one is primarily interested in the large-time behavior of solutions of the evolution equation (1.3.1). Here the attractor of the dynamical system deﬁned by the equation plays an important role. In bounded domains the attractor often consists of stationary solutions. In unbounded domains, if the equa-tion is invariant with respect to spatial translaequa-tions, the attractor may also contain traveling wave solutions.

The class of bounded solutions of (1.3.1) is particularly relevant, since most of patterns defined on infinitely extended domains correspond to its uniformly bounded solutions. We will denote this class by B. Since the mid-1990s, it has become evident that the structure of B can be very rich indeed, and includes a wealth of solutions of different nature, depending both on the function F and on the value of q. For example, consider the simplest linear equation

u00+ λu = 0, λ_{∈ R.}

In this case, the set B is very restricted: if λ < 0, it consists of the trivial solution only; if λ = 0, it consists of the constants; ﬁnally, if λ > 0, it consists of the linear combination of sin(x√λ) and cos(x√λ).

When the nonlinearity is introduced, the set B becomes richer. Below we list some of the most important type of solutions contained in B that are object of our interest. For the remaining part of this section, we suppose that the nonlinearity F has two distinct equilibria u(x) = x1,2, at the same energy level, as in the case of

equation (1.3.2).

1. Heteroclinic and homoclinic solutions. From a qualitative point of view, it is

interesting to study the connections between the equilibria by trajectories of solu-tions of the equation. These are called homoclinic or heteroclinic solusolu-tions, some-times pulses or kinks, according to whether they describe a loop based at one single equilibrium or they ”start” and ”end” at two distinct equilibria. More speciﬁcally, we say that u is a heteroclinic solution of (1.3.1) connecting x1 to x2 if

lim x→−∞(u, u 0_{, u}00_{, u}000_{)(x) = (x} 1, 0, 0, 0) and lim x→+∞(u, u 0_{, u}00_{, u}000_{)(x) = (x} 2, 0, 0, 0). (1.3.5) Of course, u is a heteroclinic solution connecting x2 to x1 if the previous relations

hold with the limes to +∞ and −∞ inverted. We say that u is a homoclinic solution of (1.3.1) if

lim x→±∞(u, u

0_{, u}00_{, u}000_{)(x) = (x}

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2. Periodic solutions. The existence of this type of solution corresponds to the

formation of spatially periodic patterns in systems described by (1.3.1). Periodic solutions can be classiﬁed according to the energy (1.3.4) or according to their period, 2T .

3. Chaotic solutions. They present an inﬁnite number of jumps between the

constant solutions and possesses between successive jumps a prescribed number of small oscillations around x1 and x2. Most of them are multibump solutions, whose

graphs have more than one critical point in a half-period if they are periodic and between tails if they converge to one or both the equilibria.

1.3.1 Linearization

The set B has been the object of much research in the last years. The reason is that it is strongly aﬀected by q, since the nature of the equilibria of F changes at the critical values of q. Therefore, this parameter has a key role in the analysis of the behavior of solutions of equation (1.3.1) (see [103, 104, 105, 106]). Suppose that

F0(0) = 0 and F00(0) = k > 0. Linearizing near zero we obtain

u0000+ qu00+ ku = 0, (1.3.6)

and so the associated characteristic equation λ4_{+ qλ}2_{+ k = 0, with eigenvalues}

λ =_±

s

−q ±pq2_{− k}

2 .

When q ≤ −√k, the four eigenvalues are real and so u = 0 is a saddle-node.

For q ∈ (−√k,√k), they are all complex with non-vanishing real parts. In this

case the equilibrium is called saddle-focus. When q = √k there is a reversible

Hopf bifurcation, and all four eigenvalues become purely imaginary. They remain imaginary for all q >√k. In this case, u = 0 is a center.

Figure 1.1: Equilibria

The behavior of the solutions of (1.3.6) provides important informations con-cerning the solutions of the nonlinear equation. For example, in the saddle-focus case, it is well known that, under suitable smoothness assumptions, the nonlinear flow and the flows defined by the linearization are conjugate in a neighborhood of the equilibrium (see [63]). Consequently, in this case, as well as the saddle-node one, the solutions of (1.3.1) inherit some properties of the solutions of (1.3.6) when they are close to u = 0. For example, we easily obtain a qualitative description of any heteroclinic at ±∞. Indeed, when 0 is a saddle-node, the solutions of (1.3.6)

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that vanish at +∞ or −∞ are monotone, while in the saddle-focus case, they do oscillate around zero.

1.4 Methods

Different methods have been developed to study equation (1.3.1). Since it contains only even order derivatives and is autonomous, this system is both reversible and Hamiltonian. This perspective then allows one to apply general results about dy-namical systems. Thus we can analyze in detail the bifurcation and structure of different periodic solutions and homoclinic orbits near critical points (see [8, 139]). Moreover, results of Devaney as well as Vanderbouwhede and Fiedler [43, 140] can be used to find families of periodic solutions on the basis of the existence of homoclinic orbits. An important restriction of these methods is that they are in some sense local, valid either near an equilibrium point or near a homoclinic or a heteroclinic orbit.

In order to derive global results, a variety of alternative approaches have been developed. Below we brieﬂy describe some of them.

1.4.1 Topological shooting

What has come to be called the shooting method has its origin in a more sophisti-cated technique, due mainly to Wa˙zewski [145]. It makes use of a topological lemma that is related, in RN_{, to Brouwer’s fixed point theorem. Shooting may be thought} of as including Wa˙zewski’s method, but also simpler topological arguments involving only connectedness (see [38, Section 2.5]). It is possible to use it more broadly, for each argument in which a boundary value problem is shown to have a solution by considering the topology of the space of initial condition. In other words, we solve a boundary value problem by reducing it to an initial value problem. The main idea of a classical shooting method is to look at the way solutions change with respect to initial conditions (taken as parameters) at some fixed initial point. Roughly speak-ing, we ’shoot’ out trajectories in different directions until we find a trajectory that has the desired boundary value. Consider for example the following boundary value

problem (see [106] for further details)

u00_{− u = 0} on (0, 1),

u(0) = 0 and u(1) = 3, (1.4.1)

and suppose that we wish to prove that there exists a solution of it. There are many ways to do this. In the method based on topological shooting, one replaces all the conditions at one of the boundary points by additional conditions at the other boundary point, so that near this point the equation has a unique solution which satisﬁes the combined old and new boundary conditions. For instance, here we can replace the condition at x = 1 and impose a slope u0 _{at x = 0. We are then left}

with the initial value problem

u00− u = 0 on (0, 1),

u(0) = 0 and u0(0) = α, (1.4.2)

where α ∈ R is a parameter which we are free to choose. By standard ODE theory [36], problem (1.4.2) has a unique local solution u(x, α), for every α ∈ R. The

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question is now whether there exists a value α∗ _{such that u( · , α}∗_{) exists on [0, 1]}

and u(1, α∗_{) = 3.}

Since the diﬀerential equation is linear, for each α ∈ R, the solution u(x, α) can be continued all the way to x = 1. Thus, u(1, α) is well deﬁned on R, and we can introduce the sets

S+= {α ∈ R : u(1, α) > 3} and S− _{= {α ∈ R : u(1, α) < 3}.}

Evidently, if α ∈ S+_{, the solution hits the line {x = 1} in the (x, u)-plane too high,}

while if α ∈ S−_{, then the solution hits the line {x = 1} too low.}

Suppose now that one has shown that α₋∈ S−_{, α}₊_{∈ S}+ _{and that the function}

Φ(α) := u(1, α) is continuous on the interval [α₋, α+]. Then plainly the sets S+

and S−_{, restricted to [α}

−, α+], are open and the existence of an α∗ ∈ [α−, α+]

where Φ(α∗_{) = 3 follows. Of course, the continuity of Φ immediately implies the}

existence of α∗_{. Notice that the previous argument says nothing about the number}

of solutions of problem (1.4.1).

In [103, 104, 106], Peletier and Troy developed a topological shooting method especially adapted to track monotone heteroclinics for the EFK equation (1.3.2). It turns out that their approach works well for q ≤ −√8, the saddle-nodes case. 1.4.2 Hamiltonian method

The evolution of many conservative systems can be described by Hamilton’s equa-tions: ˙ri = ∂H ∂pi (r, p), 1 ≤ i ≤ N, ˙pi = − ∂H ∂ri(r, p), 1 ≤ i ≤ N. (1.4.3)

Here, (r, p) belongs to RN _{× R}N_{, the so-called phase space, and N is the number} of degrees of freedom. The ﬁrst N components r = (r1, . . . , rN) represent position variables, and the last N ones p = (p1, . . . , pN) momentum variables. The function

H : RN × RN _{→ R, the Hamiltonian, represents the energy of the system. It is} an immediate consequence of equation (1.4.3) that H is an integral of motion, i.e.,

H(r(t), p(t)) is constant along any solution of (1.4.3). In other words, it holds d dt[H(r(t), p(t))] = N X i=1 ∂H ∂ri ˙ri+ N X i=1 ∂H ∂pi ˙pi = N X i=1 ∂H ∂ri ∂H ∂pi + N X i=1 ∂H ∂pi − ∂H ∂ri _{= 0,}

for every solution of (1.4.3), and so the energy is a conserved quantity.

In many problems that arise from nonlinear mechanics, as the modeling of non-linear water waves, the Hamiltonian system has an energy functional which is given by

H(r, p) = 1

2(Sp, p) + V (r), (r, p) ∈ RN× RN. (1.4.4) Here, the Hamiltonian has the classical ’kinetic plus potential’ form while ( , ) de-notes the inner product of RN_{. Moreover, suppose that}

S : RN _{→ R}N is a symmetric linear operator with eigenvalues

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This means that the quadratic form (Sp, p) is indeﬁnite but not degenerate. Thus, the Hamiltonian system is given by

˙r(t) = Sp(t),

− ˙p(t) = V0(r(t)), t∈ R. (1.4.5) Hofer and Toland [65] developed a theory for the existence of homoclinic, hetero-clinic and periodic orbits of (1.4.5), mainly based on topological methods like the antipodal mapping theory and the Brouwer degree theory. They considered only a class of Hamiltonian systems, for which the p-dependance is explicitly required to be

indefinite. In [47, 113, 114, 146, 147] periodic solutions are established for a diﬀerent

class, by chiefly applying the theory of critical points for indefinite functionals, the study of geodesics and duality theory. Further refinements of the ideas of [65] have been made in [25], yielding the existence of a family of homoclinic orbits of (1.4.5) when the potential is of the form

V (u, u00) = 1 2u2− 1 3u3− 1 2u002.

It is worth noting that equation (1.3.1) can be seen as an Hamiltonian system (1.4.5), where

r = (u, u00), p = (u000+ qu0, u0), V (r) = F (u)− u₂002, S = 0 1

1 q

!

,

and of course the Hamiltonian is given by (1.4.4). It is immediate to see that the kinetic energy is indeﬁnite, since

det −λ 1 1 q_{− λ} ! = 0 ⇔ λ2_{− λq − 1 = 0 ⇔ λ} 1,2= q_±pq2_{+ 4} 2 ,

which means that λ1< 0 for every q ∈ R. The Lagrangian associated to this system

is

L(r) = 1₂(S−1_{r, r)}_{− V (r).}

Since S is symmetric but indefinite for any q, the Hamiltonian approach seems difficult to exploit. Moreover, even if (1.3.1) can be viewed in the framework of the theory developed in [65] for such indefinite systems, the potential V is by far too general for the method to directly succeed. Instead, we take advantage of the particular structure of (1.4.5) and look at it as a higher order Lagrangian problem. See the next section for further details.

1.4.3 Variational methods

In the study of bounded stationary solutions to equation (1.3.1) on R, variational methods take up an important place. The reason is that (1.3.1) has a variational structure and is the Euler-Lagrange equation of the functional

Jq(u) = ˆ

I

Lq(u(x), u0(x), u00(x))dx, (1.4.6) where Lq is the second order Lagrangian

Lq(u, u0, u00) := 1₂

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Recall that we are used to refer to the last term of Lq as the potential. Variational methods have been widely used to establish the existence of heteroclinic, homoclinic and periodic solutions as critical points of Jq in appropriate sets of functions (see [24, 27, 31, 68, 69]). When q ≤ 0, Jq is nonnegative, so we can look for solutions of (1.3.1) as its minimizers. When q > 0, the functional is no more nonnegative, so the method of searching the minimizers may be no longer at hand. In any case, we can look for its critical points via other arguments, for instance by applying the Mountain Pass Theorem.

Solutions of (1.3.1) are critical points of Jq in diﬀerent functional spaces de-pending on the type of solution considered. For example, for homoclinic orbits satisfying

0_{, u}00_{, u}000_{)(x) = (0, 0, 0, 0),}

the appropriate space would be the Sobolev space H2_{(R). Chen and McKenna}

[31] have used this approach and employed the Mountain Pass Lemma and the Concentration Compactness Principle to prove the existence of pulses if, respectively,

f (s) = s_{− s}2_{, as in the water wave problem, and f(s) = (s + 1)}

+− 1, as in the

suspension bridge problem. For both of these problems, they considered −2pf0_{(0) <}

q < 2pf0_{(0), the range of values of q for which the origin is a saddle-focus.}

Moreover, the integral in (1.4.6) can be taken on various sets I according to the domain of the solution and the functional space on which Jq is deﬁned. For heteroclinic solutions, since they are deﬁned on R, we consider I = R, leading to

Jq(u) = ˆ R 1 2 |u00|2− q|u0|2+ F (u)dx.

This functional is well defined for functions u having first and second square inte-grable derivatives and being such that the potential F is inteinte-grable. Taking into account conditions (1.3.5), we can then define Jq in the space

{u: R → R | u − x1∈ H2(R−), u − x2∈ H2(R+)}.

The existence of heteroclinic solutions of (1.3.1) via variational arguments was in-vestigated for the ﬁrst time by Peletier, Troy and van der Vorst [107] and Kalies and van der Vorst [69]. For q ≤ 0 and F (u) = 1

4(1 − u2)2, Peletier et al. [107] proved

the existence of a minimizer of Jq in the subset of odd functions of the space E = {u: R → R | u(0) = 0, u + 1 ∈ H2(R−_{), u − 1 ∈ H}2_(R+_)}.

To obtain an odd heteroclinic solution it is suﬃcient to look for critical points of the functional J_q+(u) := ˆ R+ 1 2 |u00|2− q|u0|2+1 4(u2− 1)2dx in the space E+:= {u | u − 1 ∈ H2_(R+_{), u(0) = 0}.}

Indeed, the condition u00_{(0) = 0 is a natural boundary condition fulﬁlled by any}

crit-ical point of J+

q in E+so that their odd extensions solve the Euler-Lagrange equation (1.3.2) on R. These methods have been considerably reﬁned in [68, 69]. Of course, the previous arguments extend easily to a functional deﬁned from any second order positive Lagrangian with a symmetric potential having two non-degenerate minima

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(with non-vanishing second derivative) at the same energy level and superquadratic grows at ±∞. When we look for T -periodic solutions, the natural space we consider is the real Hilbert space

HT := {u : u ∈ H2([0, T ]), u0 ∈ H01([0, T ])},

with scalar product

hu, viHT = ˆ T 0 u00v00dx + ˆ T 0 uvdx

and corresponding norm kukHT. Of course the action functional is deﬁned by

Jq,T(u) := ˆ T 0 1 2 |u00|2− q|u0|2+ F (u)dx,

while the dependance on T is often omitted when there is no risk of confusion.

1.5 Some results from the literature

As already said, the nature of equation (1.3.1) depends both on the choice of the potential F and on the sign of the parameter q and, of course, the existence results reﬂect this inﬂuence. We begin with some results involving a double-well potential, and after we will deal with a single-well coercive potential.

1.5.1 _{F is a double-well potential} In the model case F (u) = 1

4(1 − u2)2, equation (1.3.1), as already seen, becomes

u0000+ qu00_{+ u}3_{− u = 0.} _(1.5.1)

Peletier and Troy [106] gave a complete catalogue of the set B of bounded solutions of equation (1.5.1) for the parameter range q ∈ (−∞, −√8]. Taking into account Section 1.3.1, this is the saddle-nodes case, when the spectrum at the stable uniform states u = +1 and u = −1 is real valued. Their results can be summarized as follows.

Theorem 1.5.1 (Theorems 2.1.1-2.2.1 of [106]). Let q ∈ (−∞, −√8]. Then, the

following facts hold true:

1. For every E _{∈ (0,}1₄) there exists a periodic solution uE to equation (1.5.1),

which is even with respect to its critical points, odd with respect to its zeros, and has the bound

kuEk∞<

q

1 − 2√E.

2. There exists an odd monotone heteroclinic solution of equation (1.5.1) that satisfies

0_{, u}00_{, u}000_{)(x) = (±1, 0, 0, 0).}

They obtained these results by using the method of topological shooting in-troduced in Section 1.4.1. Moreover, they proved that the solutions obtained in Theorem 1.5.1 are the only bounded non-constant solutions of equation (1.5.1) for these values of q.

When q > −√8, the spectrum at the equilibria u = ±1 is complex valued. An im-mediate consequence is that, in this parameter regime, homoclinic and heteroclinic

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solutions leading to either of these two constant solutions cannot be monotone. In fact, as q passes through −√8, the set B instantly becomes much richer and the solution graphs more complex (see [25, 27, 42] and the references therein). In this range, the linearization around the equilibria displays oscillatory solutions so that any heteroclinic of (1.5.1) oscillates around ±1 in its tails, i.e., when x → ±∞. This oscillatory behavior close to the equilibria makes a shooting method much more tedious, since one of the greatest diﬃculties is to control the convergence at inﬁnity. However, Peletier and Troy adapted their arguments in [104], and after a careful analysis they managed to single out two families of odd heteroclinics in the range

q ∈ (−√8, 0], which differ by the amplitude of the oscillations. The first one con-sists of the so-called multi-transition solutions, since all the successive local extrema between the zeros rely outside the region [−1, 1]. It contains, for each n ∈ N, a solution whose profile displays 2n + 1 jumps from −1 to +1 and two oscillatory tails around −1 and +1. The second family contains the single-transition heteroclinics, whose solutions display oscillations with an amplitude smaller than 1. They may also be classified according to their number of oscillations around 0. Both these families have a similar structure, which can be divided into three different regions: an inner region (−L, L), where the solution oscillates around u = 0, and two outer regions, (−∞, −L) and (L, +∞), which contain the tails where the solution in the inner region joins up with one of the stable uniform states ±1.

Moreover, Kalies and van der Vorst [69] constructed the so-called multi-bump solutions, which are characterized by multiple oscillations separated by large dis-tances. The usual methods to obtain such solutions are rather tricky and require a careful study of the stable and unstable manifolds (see [27, 37, 120]). Kalies et al. [68] introduced a direct method to find multi-transition solutions. Note that such solutions are qualitatively different from multi-bump ones, as the distance between two successive transitions is not necessarily large. The method in [68] consists in minimizing the action functional Jq in specific subspaces of functions having a com-mon homotopy type. Basically, the homotopy type describes the trajectory of any function in the uu0_{-plane by recording the number of transitions from one}

equilib-rium to the other and counting the number of turns it makes around −1 and +1 in between the transitions. Their method perfectly handles oscillatory graphs and is therefore eﬃcient when −√8 ≤ q ≤ 0.

Existence results in the parameter range q ≤ 0 can be also obtained through diﬀerent methods. For example, Peletier et al. proved the existence of heteroclinic solutions to equation (1.5.1) via variational argument, see [107].

The dynamics of equation (1.5.1) with q > 0 is much less understood than the EFK case. Numerical experiments (see [15]) suggest that a large variety of those solutions found for q ≤ 0 still exists for a certain range of positive values of q. However, the limitations of the shooting method of Peletier and Troy was pointed out by van den Berg [14].

As q becomes larger than −√8 (in particular positive), a multitude of periodic solutions with different structures emerges, as described in [106]. We pay specific at-tention to two families, that consist of odd and even periodic solutions, respectively, each with zero energy. Observe that they are only a part of the set of periodic solu-tions that exist in this parameter range. The first family consists in two branches of single-bump periodic solutions (with a single oscillation), which emerge at the value

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amplitude smaller than 1. The second one consists in a countable pairs of family of branches: they are divided into Γna and Γnb, whose solutions are convex and concave at the origin, respectively. Both of these families extend over the intervals (−√8, qn), where qn= √ 2 n + 1 n , n = 2, 3, . . .

The following is an existence result of single-bump periodic solutions.

Theorem 1.5.2 _{(Theorem 4.1.1. of [106]). For every q > −}√8 there exist

single-bump periodic solutions u₋ and u+ of equation (1.5.1) such that E(u±) = 0 and

ku−k∞< 1 and ku+k∞> 1.

Moreover, the functions u_± are odd with respect to their zeros and even with respect to their critical points.

There is also a family of odd multi-bump periodic solutions with the character-istic property that the maxima all lie above u = 1 and the minima all lie below

u = _{−1, with the exception of the first point of symmetry, ζ in R}+, where the

situation is reversed (see [106, Theorem 4.1.3]).

We conclude this section with three qualitative results. The ﬁrst one is a sharp universal upper bound for bounded solutions of equation (1.5.1) when q ≤ 0, while the other two describe the asymptotic behavior of bounded solutions when q ∈ (−√8, +∞).

Theorem 1.5.3 (Lemma 2.4.3 and Lemma 2.4.5 of [106]). If q ≤ 0, then any

bounded solution u of equation (1.5.1) satisfies

|u(x)| <√2, ∀ x ∈ R.

When, in particular, q≤ −√8, then

|u(x)| < 1, ∀ x ∈ R.

For the next results, we denote by ϕ the odd increasing kink at q = −√8.

Theorem 1.5.4 (Theorem 4.2.2 of [106]). Let (qn) ⊆ (− √

8, +∞) be a decreasing

sequence such that qn → −√8 as n → +∞, and let un(x) = u(x, qn) be a

corre-sponding sequence of odd solutions of equation (1.5.1) with zero energy and such that u0_n(0) > 0. Then,

un→ ϕ as n → +∞

uniformly on bounded intervals.

Theorem 1.5.5 (Lemma 4.2.4 of [106]). Let u(x, q) be an odd periodic solution of

equation (1.5.1) with zero energy, symmetric with respect to its critical points and such that _{ku( · , q)k}_∞< 1. Then,

ku(q)k∞< 1

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1.5.2 _{F is a single-well coercive potential}

When F is a convex, coercive potential, the study of (1.3.1) became a major tool to understand the modeling of suspension bridges, see [77, 95, 96]. When we search for traveling waves that decay to zero exponentially as |x| → ∞, we are ultimately concerned with ﬁnding homoclinic solutions of the nonlinear ordinary diﬀerential equation

y0000+ c2_y00_{+ (1 + y)}

+− 1 = 0. (1.5.2)

The idea in [96] was to write the analytic expressions for the solutions of y0000₊

c2y00+ y = 0 for y ≥ −1 and y0000 + c2y00_{− 1 = 0 for y ≤ −1, and then ensure}

the continuity of these solutions and their ﬁrst three derivatives whenever y = −1. However there were some problems with this approach. Indeed, the existence was not proved rigorously since all calculations were approximate. Nonetheless, the paper led to some obvious conjectures: it seemed that the number of solutions could be quite large and, moreover, the L∞_{-norm of the solutions seemed to go to +∞,}

as c → 0.

Furthermore, the method of ﬁnding traveling wave solutions was heavily depen-dent on the analytic form of the nonlinearity in (1.5.2). Thus, the publication of this paper left open several interesting questions. First, could one verify that, as

c→ 0, the L∞_{-norm of the solutions goes to +∞, as was indicated by the}

computa-tions? Second, could one prove existence (and multiplicity) for solutions of a more general nonlinearity with the same basic shape as that of (1.5.2)? And third, which are the stability, instability and interaction properties of these traveling waves? In [79], Lazer and McKenna gave a result of non-existence about what happens for the parameter value c = 0.

Theorem 1.5.6 (Theorem 1 of [79]). The solution u ≡ 0 is the only solution of the

equation

y0000+ (1 + y)+− 1 = 0

such that _kuk_∞ is bounded.

In order to give to the reader the idea of what it is known so far in the literature, below we list some results about the argument. Note that the hypotheses on F have been further specialized, leading to consider coercive and quasi-convex potentials, i.e., those satisfying

F0(t)t ≥ 0, ∀ t ∈ R. (1.5.3)

Existence results

Let us ﬁrst remark that, for coercive potentials, condition (1.5.3) has proven to be almost equivalent to the absence of (nontrivial) bounded solutions to (1.3.1). Here there is a more precise statement.

Theorem 1.5.7_{(Theorems 3.1 and 5.1 in [97]). Let q ≤ 0. If (1.5.3) holds, then the}

only bounded solutions of (1.3.1) are constants. Moreover, in the class of coercive potentials with _{F0(t) = 0} discrete, (1.5.3) is actually equivalent to the existence of bounded nontrivial solutions.

Under assumption (1.5.3), one may still seek for global unbounded solutions. An immediate ODE argument shows that, if F0 _{is globally Lipschitz, all local}

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However, the peculiar nature of (1.3.1) allows the following one-sided generalization (notice that this holds for any q ∈ R).

Theorem 1.5.8 (Theorem 1 in [13]). Let q ∈ R be arbitrary and F ∈ C2 satisfy F0(t)t > 0, for all t 6= 0. If

either lim sup

t→+∞

F0(t)

t < +∞ or lim sup_t→−∞ F0(t)

t < +∞, (1.5.4) then any solution to (1.3.1) is globally defined.

Regarding non-existence, Gazzola and Karageorgis proved the following. Recall that with E we mean the Hamiltonian energy (1.3.3).

Theorem 1.5.9 _{(Theorem 3 in [53]). Let q ≤ 0. Suppose F is a convex potential}

satisfying

F (0) = 0, F0(t)t ≥ c|t|2+ε for ε > 0, F0(t)t ≥ cF (t) ∀ |t| >> 1 for some c > 0 and

lim inf

|t|→+∞

F (λt) F (t)α > 0

for some λ_{∈ ]0, 1[, α > 0. If u solves (1.3.1) in a neighborhood of 0 and}

either u0(0)u00_{(0) − u(0)u}000_{(0) − qu(0)u}0_{(0) 6= 0} _or _{E 6= 0,} _(1.5.5)

then u blows up in finite time.

As we will see, the situation for q > 0 is more complex. Regarding non-existence of nontrivial solutions, the seemingly most up-to date results are the following.

Theorem 1.5.10 _{(Theorem 1 in [115]). Let F ∈ C}2 _satisfy

a|t|p+1≤ F0(t)t ≤ b|t|r+1_{+ c|t|}p+1_, _{for some a, b, c > 0 and 1}_{≤ r < p . (1.5.6)}

Then, for any q > 02, there exists E0= E0(a, b, c, p, r, q) ≥ 0 such that any solution

to (1.3.1), satisfying _{E(u) > E}0, blows up in finite time.

Theorem 1.5.11 (Theorem 1 in [49]). Let F ∈ C2 satisfy (1.5.6) and

F00(t) > F00(0), for all t_{6= 0.} (1.5.7) If q > 0 satisfies q2 _{≤ 4F}00_{(0), then the only globally defined solution to (1.3.1) is}

u≡ 0.

Still in [49], the rôle of the condition q < 2pF00_{(0) is also discussed, through the}

following partial converse of Theorem 1.5.11.

Theorem 1.5.12 _{(Theorem 2 in [49]). Suppose F ∈ C}2 _{is even, satisfies (1.5.6),}

(1.5.7) and the limit lim_t→+∞F0_(t)

tp exists. Then, for every q > 0 such that q2 >

4F00_{(0), there exists a nontrivial periodic solution to (1.3.1).}

Notice that, being p > 1, (1.5.6) forces lim t→+∞

F0(t)

tp = B > 0, which implies the (much weaker) condition

lim inf

|t|→+∞

F (t)

t2 = +∞.

2_{In [115], this theorem is actually proved for 0 < q ≤ 2, but a simple scaling argument shows} its validity for any q > 0. Indeed, if u solves (1.3.1), then uλ(x) := u(x/

√

λ) solves u0000λ + λqu00λ+

Fλ0(uλ) = 0, where Fλ(t) := λ2F (t) satisfies (1.5.6) with the same exponents and with constants

(28)

Asymptotic behavior

In light of the previous discussion, under assumption (1.5.3) it only makes sense to consider the asymptotic behavior of the solutions to (1.3.1) for q ↓ 0. The starting point is a result proved by Lazer and McKenna.

Theorem 1.5.13 (Theorem 2 in [79]). Let, for some qn↓ 0, (un) be a sequence of

bounded nontrivial solutions to

u0000+ qnu00+ (1 + u)+− 1 = 0.

Then, kunk∞→ +∞.

We brieﬂy say that the nontrivial solutions to (1.3.1) are unbounded as q ↓ 0 if the thesis of the previous theorem holds for any qn↓ 0 and corresponding nontrivial solutions (un) to (1.3.1). The previous result has later been generalized as follows.

Theorem 1.5.14 _{(Theorem 3.2 in [97]). Let F ∈ C}2 _{satisfy F}00_{(0) > 0, int({F}0 ₌

0}) = ∅ and (1.5.3). Then, the nontrivial solutions to (1.3.1) are unbounded, as

q_{↓ 0.}

The condition int({F0 _{= 0}) = ∅ is readily seen to be necessary for the thesis,}

as the following remark shows.

Remark 1.5.1 (Remark 3.2 in [97]). Suppose that [a, b] ⊆ {F0 = 0}, then uβ(x) :=

A sin(βx) + B with A = (b_{− a)/4, B = (a + b)/2 solves (1.3.1) for each β, being} uniformly bounded.

1.6 Our results

In [92] we gave some answers to the questions raised by Lazer and McKenna in [79], considering equation (1.3.1) for coercive, quasi-convex potentials F . For the reader’s convenience, below we recall that (1.3.1) reads as

u0000+ qu00+ F0(u) = 0. (1.6.1)

Notice that some of our statements will involve quantities like lim sup t→±∞ F (t) t2 , lim inf_t→±∞ F (t) t2 .

Similar, but weaker, statements will hold under similar conditions on f(t) = F0_(t),

namely involving the corresponding quantites lim sup t→±∞ f (t) t , lim inft→±∞ f (t) t

(which are more frequent in the literature), simply due to the inequalities lim sup t→±∞ F (t) t2 ≤ lim sup t→±∞ f (t) t , lim inft→±∞ F (t) t2 ≥ lim inf_t→±∞ f (t) t ,

A-priori estimates for some classes of elliptic problems